On operators with the same local spectra

Aleksandar Torgašev

Czechoslovak Mathematical Journal (1998)

  • Volume: 48, Issue: 1, page 77-83
  • ISSN: 0011-4642

Abstract

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Let B ( X ) be the algebra of all bounded linear operators in a complex Banach space X . We consider operators T 1 , T 2 B ( X ) satisfying the relation σ T 1 ( x ) = σ T 2 ( x ) for any vector x X , where σ T ( x ) denotes the local spectrum of T B ( X ) at the point x X . We say then that T 1 and T 2 have the same local spectra. We prove that then, under some conditions, T 1 - T 2 is a quasinilpotent operator, that is ( T 1 - T 2 ) n 1 / n 0 as n . Without these conditions, we describe the operators with the same local spectra only in some particular cases.

How to cite

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Torgašev, Aleksandar. "On operators with the same local spectra." Czechoslovak Mathematical Journal 48.1 (1998): 77-83. <http://eudml.org/doc/30403>.

@article{Torgašev1998,
abstract = {Let $B(X)$ be the algebra of all bounded linear operators in a complex Banach space $X$. We consider operators $T_1,T_2\in B(X)$ satisfying the relation $\sigma _\{T_1\}(x) = \sigma _\{T_2\}(x)$ for any vector $x\in X$, where $\sigma _T(x)$ denotes the local spectrum of $T\in B(X)$ at the point $x\in X$. We say then that $T_1$ and $T_2$ have the same local spectra. We prove that then, under some conditions, $T_1 - T_2$ is a quasinilpotent operator, that is $\Vert (T_1 - T_2)^n\Vert ^\{1/n\} \rightarrow 0$ as $n \rightarrow \infty $. Without these conditions, we describe the operators with the same local spectra only in some particular cases.},
author = {Torgašev, Aleksandar},
journal = {Czechoslovak Mathematical Journal},
keywords = {Banach space; spectrum; local spectrum; Banach space; spectrum; local spectrum},
language = {eng},
number = {1},
pages = {77-83},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On operators with the same local spectra},
url = {http://eudml.org/doc/30403},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Torgašev, Aleksandar
TI - On operators with the same local spectra
JO - Czechoslovak Mathematical Journal
PY - 1998
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 48
IS - 1
SP - 77
EP - 83
AB - Let $B(X)$ be the algebra of all bounded linear operators in a complex Banach space $X$. We consider operators $T_1,T_2\in B(X)$ satisfying the relation $\sigma _{T_1}(x) = \sigma _{T_2}(x)$ for any vector $x\in X$, where $\sigma _T(x)$ denotes the local spectrum of $T\in B(X)$ at the point $x\in X$. We say then that $T_1$ and $T_2$ have the same local spectra. We prove that then, under some conditions, $T_1 - T_2$ is a quasinilpotent operator, that is $\Vert (T_1 - T_2)^n\Vert ^{1/n} \rightarrow 0$ as $n \rightarrow \infty $. Without these conditions, we describe the operators with the same local spectra only in some particular cases.
LA - eng
KW - Banach space; spectrum; local spectrum; Banach space; spectrum; local spectrum
UR - http://eudml.org/doc/30403
ER -

References

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  1. Quasinilpotent equivalence of not necessarily commuting operators, Journal Math. Mech. 15 (1966), 521–540. (1966) MR0192344
  2. Theory of Generalized Spectral Operators, Gordon and Breach, New York, 1968. (1968) MR0394282
  3. 10.1007/BF01234965, Arch. Math. 14 (1963), 341–349. (1963) MR0152893DOI10.1007/BF01234965
  4. 10.2140/pjm.1968.27.305, Pacific J. Math. 27(2) (1968), 305–324. (1968) Zbl0172.17204MR0236738DOI10.2140/pjm.1968.27.305
  5. 10.2140/pjm.1964.14.333, Pacific J. Math. 14 (1964), 333–352. (1964) Zbl0197.39501MR0164242DOI10.2140/pjm.1964.14.333
  6. On local spectral properties of operators in Banach spaces, Czechoslovak Math. J. 23 (1973), 483–492. (1973) MR0322536

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